On Quantum Wave Groups

G.R.Dixon, 3/9/2007

In this article the "spinning spiral" model for the complex quantum wave function Y is investigated. The associated particle is a free electron moving in the positive x direction at a most probable speed of v_{o}. At any given x a complex plane is assumed to exist perpendicular to the x-axis. The Imaginary axis of this plane is parallel to the y-axis, and the Real axis is parallel to the z-axis. Representations of Y in any of these complex planes will be looking down the x-axis in the positive x direction (See Fig. 1_1).

Figure 1_1

Complex Plane Parallel to the yz Plane

The electron is considered to be localized which, according to the Heisenberg Uncertainty Principle, implies that there is a finite probability it could be found to have any of a range of momenta p. The localization is defined by

. (1_1)

For purposes of computation, the momenta are assumed to be discrete, say p_{i} where i ranges from 1 to N. Each momentum in the set has an associated Y_{i} with its own wave number (k_{i}) and angular frequency (w_{i}).

As suggested in a previous
article, each Y_{i} __spins__ CW with increasing time and __spirals__ CCW with increasing x. The angular rate at which Y_{i} spins at any given x is defined by the deBroglie relation

. (1_2)

And the rate at which Y_{i} spirals at any given time is defined by

. (1_3)

For the localized particle, Y is the sum of the set of Y_{i}. The magnitude of each Y_{i} is 1/N, thus assuring that the magnitude of the resultant Y is never greater than unity. It is noteworthy that the individual Y_{i} need not be weighted in order to obtain a wave packet (although this is often done for purposes of fourier analysis, etc.). For graphic purposes, YY* (which is always real) is plotted vs. x. The general goal is to determine (a) if the Y_{i} sum to a quasi-gaussian group, and (b) if the group velocity equals v_{o} (the most probable velocity).

2. The Ranges of p and x.

We assume that, at t=0, all the constituent Y_{i} __are Real__ and __are in phase at x=0__. Since the wave numbers vary, the constituent Y_{i} will fall out of phase as we move away from the origin. The uncertainty in x is assumed to be Dx=1E-6 meters (wavelength of visible light). The most probable momentum is specified to be p_{o}=.01mc, where m is the electron’s mass and c is the speed of light. The uncertainty in p is Dp=h/Dx, and p ranges from p_{o}-Dp/2 to p_{o}+Dp/2.

3. Plots of YY*.

We begin by computing the resultant Y at time t=0 and over the range -2Dx__<__x__<__4Dx. Fig. 3_1 plots YY*(x) at t=0. Note the well-defined group and (since N is finite) the side lobes.

Fig 3_1

YY*(x), t=0

Fig. 3_2 plots YY*(x) for t=2Dx/v_{o}. Note how the group has propagated to the right a distance of v_{o}t. That is, the constituent Y_{i} are now all in phase at x= v_{o}t.

Figure 3_3

YY*(x), t=2Dx/v_{o}